Sauron did not directly participate in the creation of the Three Rings, so they were free of its corrupting influence.
So although he did intend the One Ring to rule them all, the Three were tied to the One's power but their wearers did not become corrupted by Sauron's influence. Thus there was no compelling need to destroy them. This leaves the question of why the Three were kept, rather than discarded or destroyed just in case.
I believe the Three's power after the destruction on the One is left unresolved. During the council of Elrond (LOTR 2.II), the assembled experts don't know:
‘But what then would happen, if the Ruling Ring were destroyed, as you counsel?’ asked Glóin.
‘We know not for certain,’ answered Elrond sadly. ‘Some hope that the Three Rings, which Sauron has never touched, would then become free, and their rulers might heal the hurts of the world that he has wrought. But maybe when the One has gone, the Three will fail, and many fair things will fade and be forgotten. That is my belief.’
Galadriel seems to hold a similar opinion (LOTR 2.VII).
[Galadriel is talking to Frodo.] ‘Yet if you succeed, then our power is disminished, and Lothlórien will fade, and the tides of Time will sweep it away. We must depart into the West, or dwindle to a rustic folk of dell and cave, slowly to forget and to be forgotten.’
After the destruction of the One Ring, Gandalf has this to say (LOTR 6.V):
‘The Third Age of the world is ended, and the new age is begun […]; the power of the Three Rings is also ended. And all the lands that you see, and those that lie about them, shall be dwellings of Men. For the time comes of the Dominion of Men, and the Elder Kindred shall fade or depart.’
Finally (LOTR 6.IX), it is revealed that the Three will be worn by their respective wielders as they leave Middle-earth together:
Upon [Elrond's] finger was a ring of gold with a great blue stone, Vilya, mightiest of the Three. On [Galadriel's] finger was Nenya, the ring wrought of mithril, that bore a single white stone flickering like a frosty star.
[…] Gandalf now wore openly upon his hand the Third Ring, Narya the Great, and the stone upon it was red as fire.
The Silmarillion includes a chapter “On the Rings of Power and the Third Age”. Its summary of the events is that
[…] the One and the Seven and the Nine are destroyed; and the Three have passed away, and with them the Third Age is ended […].
But when all [the events of LOTR] were done, and the Heir of Isildur had taken up the lordship of Men, and the dominion of the West had passed to him, then it was made plain that the power of the Three Rings also was ended, and to the Firstborn the world drew old and grey.
In light of these clues, there are several reasonable theories:
- The Three have become pretty but useless baubles, kept as souvenirs.
- The Three's power is disminished but not extinguished.
- The Three's power remains intact.
My interpretation is that Tolkien purposefully left this unresolved. After the end of the Third Age, the world belongs to the humans. The magic has left. This goes beyond the Three Rings: the elves had magic before the rings and will still have magic after, but their time has passed. Maybe the Three still have some power, maybe they don't, but primarily they must not leave a mark any longer: the Three must not remain, by need of historicity. The fate of the Three Rings after they have left Middle-earth is for us humans to speculate about, not know of.